# Properties

 Label 336.6.q Level $336$ Weight $6$ Character orbit 336.q Rep. character $\chi_{336}(193,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $80$ Newform subspaces $14$ Sturm bound $384$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 336.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$14$$ Sturm bound: $$384$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(336, [\chi])$$.

Total New Old
Modular forms 664 80 584
Cusp forms 616 80 536
Eisenstein series 48 0 48

## Trace form

 $$80 q - 18 q^{3} + 62 q^{7} - 3240 q^{9} + O(q^{10})$$ $$80 q - 18 q^{3} + 62 q^{7} - 3240 q^{9} - 604 q^{11} - 2714 q^{19} - 2992 q^{23} - 26364 q^{25} + 2916 q^{27} + 8144 q^{29} + 10990 q^{31} + 4716 q^{33} - 35964 q^{35} + 10648 q^{37} + 10818 q^{39} + 19280 q^{41} - 22828 q^{43} + 32628 q^{47} + 9008 q^{49} + 7216 q^{53} - 96824 q^{55} + 12888 q^{57} - 14480 q^{59} + 12568 q^{61} + 5022 q^{63} + 42200 q^{65} + 87882 q^{67} - 272232 q^{71} - 19716 q^{73} - 78750 q^{75} - 57368 q^{77} + 18546 q^{79} - 262440 q^{81} - 208344 q^{83} - 99536 q^{85} + 90828 q^{87} - 95920 q^{89} - 265210 q^{91} - 792 q^{93} - 132516 q^{95} - 213464 q^{97} + 97848 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(336, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
336.6.q.a $2$ $53.889$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-9$$ $$-86$$ $$-49$$ $$q+(-9+9\zeta_{6})q^{3}-86\zeta_{6}q^{5}+(-98+\cdots)q^{7}+\cdots$$
336.6.q.b $2$ $53.889$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-9$$ $$-11$$ $$-259$$ $$q+(-9+9\zeta_{6})q^{3}-11\zeta_{6}q^{5}+(-133+\cdots)q^{7}+\cdots$$
336.6.q.c $2$ $53.889$ $$\Q(\sqrt{-3})$$ None $$0$$ $$9$$ $$6$$ $$-119$$ $$q+(9-9\zeta_{6})q^{3}+6\zeta_{6}q^{5}+(-126+133\zeta_{6})q^{7}+\cdots$$
336.6.q.d $2$ $53.889$ $$\Q(\sqrt{-3})$$ None $$0$$ $$9$$ $$69$$ $$-245$$ $$q+(9-9\zeta_{6})q^{3}+69\zeta_{6}q^{5}+(-147+\cdots)q^{7}+\cdots$$
336.6.q.e $4$ $53.889$ $$\Q(\sqrt{-3}, \sqrt{-83})$$ None $$0$$ $$-18$$ $$33$$ $$350$$ $$q+9\beta _{1}q^{3}+(20+20\beta _{1}+7\beta _{3})q^{5}+\cdots$$
336.6.q.f $4$ $53.889$ $$\Q(\sqrt{-3}, \sqrt{9601})$$ None $$0$$ $$-18$$ $$53$$ $$-6$$ $$q+(-9+9\beta _{2})q^{3}+(\beta _{1}+26\beta _{2})q^{5}+\cdots$$
336.6.q.g $4$ $53.889$ $$\Q(\sqrt{-3}, \sqrt{7081})$$ None $$0$$ $$18$$ $$-47$$ $$174$$ $$q+(9-9\beta _{2})q^{3}+(\beta _{1}-24\beta _{2})q^{5}+(73+\cdots)q^{7}+\cdots$$
336.6.q.h $4$ $53.889$ $$\Q(\sqrt{-3}, \sqrt{505})$$ None $$0$$ $$18$$ $$-17$$ $$408$$ $$q+9\beta _{1}q^{3}+(-8+7\beta _{1}+2\beta _{2}-3\beta _{3})q^{5}+\cdots$$
336.6.q.i $8$ $53.889$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-36$$ $$0$$ $$42$$ $$q+9\beta _{1}q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(-10+\cdots)q^{7}+\cdots$$
336.6.q.j $8$ $53.889$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$36$$ $$0$$ $$-258$$ $$q+(9-9\beta _{1})q^{3}+(\beta _{4}-\beta _{6})q^{5}+(-9+\cdots)q^{7}+\cdots$$
336.6.q.k $8$ $53.889$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$36$$ $$64$$ $$-42$$ $$q+(9-9\beta _{1})q^{3}+(2^{4}\beta _{1}+\beta _{2})q^{5}+(-22+\cdots)q^{7}+\cdots$$
336.6.q.l $10$ $53.889$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$-45$$ $$-6$$ $$97$$ $$q-9\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{6})q^{5}+(14+\cdots)q^{7}+\cdots$$
336.6.q.m $10$ $53.889$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$0$$ $$45$$ $$-75$$ $$113$$ $$q+(9+9\beta _{1})q^{3}+(15\beta _{1}+\beta _{2}-\beta _{4})q^{5}+\cdots$$
336.6.q.n $12$ $53.889$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-54$$ $$17$$ $$-144$$ $$q+(-9+9\beta _{1})q^{3}+(3\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(336, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(336, [\chi]) \simeq$$ $$S_{6}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$